Finite Group Databasebeta

← #10693057

Subgroups — #10693057

Order 768.

602 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2535C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic, complemented
3#71617C211yesmaximal, characteristic, complemented
4#71598C212yesmaximal, complemented
5#67631C212yesmaximal, complemented
6#9489—31nomaximal, Sylow
7C2×C3×#121—21no—
8C2×C3×C4×D4C2211yescharacteristic
9C2×C3×C4×D4C2211yescharacteristic
10C3×#397C412yes—
11C3×#392C412yes—
12C3×#386—22no—
13C3×#382C2211yescharacteristic
14C3×#380—21no—
15C2×#833—21no—
16#5207—22no—
17#5012—24no—
18#4925C2212yes—
19#4924C2212yes—
20D4⋊(C3⋊C8)—24no—
21#4530—21no—
22#2975—31no—
23#2535D311yescharacteristic
24#1800—31no—
25#1486—32no—
26#1286—32no—
27C22×C3×D4—22no—
28C22×C3×D4C2×C411yescharacteristic
29C22×C3×D4D411yescharacteristic
30C23×C3×C4—21no—
31C23×C3×C4D412yes—
32C3×#121—44no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4C2×C412yes—
44C2×C3×C42C2311yescharacteristic
45C2×#208—24no—
46C2×#208—22no—
47C3×#95—22no—
48C3×#95—21no—
49C3×#95C2×C412yes—
50#833—42no—
51#830—24no—
52#830—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8D414yescomplemented
57C2×#121—21no—
58C2×C4×D4C2×D311yescharacteristic
59C2×C4×D4C2×D311yescharacteristic
60#436—64no—
61#431—32no—
62#423—32no—
63C2×#100—61no—
64#407—62no—
65#397Dic312yes—
66#392Dic312yes—
67#386—22no—
68#382C2×D311yescharacteristic
69#380—21no—
70#343—61no—
71D4⋊C8—64no—
72C24×C3—21no—
73C24×C3C22⋊C412yes—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—21no—
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4—42no—
80C2×C3×D4—44no—
81C2×C3×D4—44no—
82C2×C3×D4—82no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—21no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—82no—
92C22×C3×C4C22⋊C412yes—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C3×C4⋊C4—22no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—24no—
100C3×C22⋊C4—44no—
101C3×C22⋊C4—41no—
102C3×C22⋊C4—41no—
103C3×C22⋊C4—82no—
104C3×C42C2×D412yes—
105C3×C42#4112yes—
106#208—42no—
107#208—42no—
108#208—84no—
109C2×C3⋊C8—24no—
110C2×C3⋊C8—42no—
111C22×D4—22no—
112C22×D4C2×Dic311yescharacteristic
113C22×D4#6711yescharacteristic
114C23×C4—21no—
115C23×C4#6712yes—
116C2×M16—64no—
117C2×M16—62no—
118#121—44no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4C2×Dic311yescharacteristic
125C2×C4⋊C4#6711yescharacteristic
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4C2×Dic312yes—
130C2×C42C22×D311yescharacteristic
131C4⋊C8—62no—
132C4⋊C8—62no—
133C4⋊C8—62no—
134#100—122no—
135C22⋊C8—64no—
136C22⋊C8—62no—
137C4×C8—34no—
138#95—22no—
139#95—21no—
140#95C2×Dic312yes—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—22no—
145C23×C3—42no—
146C23×C3—42no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3C2×C22⋊C411yescharacteristic
154C3×D4—24no—
155C3×D4—42no—
156C3×D4—42no—
157C3×D4—42no—
158C3×D4—88no—
159C2×C3×C4—22no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—44no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—42no—
170C2×C3×C4—82no—
171C2×C3×C4—82no—
172C2×C3×C4—82no—
173C2×C3×C4C4×D412yes—
174C2×C3×C4#12011yescharacteristic
175C2×C3×C4#12112yes—
176C2×C3×C4#12311yescharacteristic
177C2×C3×C4AGL(1,ℤ/8)12yes—
178C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
179C2×C3×C4#13711yescharacteristic
180C3⋊C8—44no—
181C3⋊C8—82no—
182C24—21no—
183C24#21712yes—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—82no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—82no—
202C22×C4#21712yes—
203C22×C4C22×Dic311yescharacteristic
204C22×C4C2×#6711yescharacteristic
205C22×C4C2×#6711yescharacteristic
206M16—122no—
207M16—122no—
208M16—244no—
209C2×C8—64no—
210C2×C8—122no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C22⋊C4—24no—
215C22⋊C4—44no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—82no—
219C42D3×D412yes—
220C42#23712yes—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—42no—
224C22×C3—44no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3—82no—
231C22×C3—81no—
232C22×C3—81no—
233C22×C3—81no—
234C22×C3—81no—
235C22×C3#38611yescharacteristic
236C22×C3#41711yescharacteristic
237C22×C3#41811yescharacteristic
238C3×C4—22no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—42no—
242C3×C4—82no—
243C3×C4—81no—
244C3×C4#46812yes—
245C3×C4#46912yes—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C2×#21711yescharacteristic
259D4—24no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—88no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#93212yes—
279C2×C4D4×Dic312yes—
280C2×C4#93511yescharacteristic
281C2×C4#93711yescharacteristic
282C2×C4#93912yes—
283C2×C4#94911yescharacteristic
284C2×C4#95111yescharacteristic
285C8—124no—
286C8—242no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—82no—
290C2×C3—82no—
291C2×C3#189111yescharacteristic
292C2×C3#189511yescharacteristic
293C2×C3#191111yescharacteristic
294C22—21no—
295C22—21no—
296C22—42no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22#521511yescharacteristic
309C22#522011yescharacteristic
310C22#523211yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—82no—
316C4—81no—
317C4#503912yes—
318C4#504212yes—
319C3#948911yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2—82no—
323C2—82no—
324C2#7168211yescharacteristic
325C2#7171411yescharacteristic
326C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2535C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#2535D3
hypocenterC3#9489
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #82304C2semidirect
3. #71617C2semidirect
4. #71598C2semidirect
5. #67631C2semidirect
56. C4×C3⋊C8D4semidirect
319. C3#9489semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#25352
#823042
#716172
#715982
#676312
#94893